Many years later I remembered my old question and started looking it up. It turns out that eight is the sum of 1+3+3+1 perhaps similarly to what's in the article.
Spherical harmonics ends up giving rise to a three dimensional 'overtone' series (borrowing from my understanding of music theory). In the first order there's only one mode of vibration. In the second order there are three additional modes. The summands above are something like positive and negative degrees of freedom for each mode in the second shell.
Here's a diagram of what the modes look like in each order:
https://i.ytimg.com/vi/OkDYbIhisZE/maxresdefault.jpg
...and here's an animation of a sphere undergoing the differing modes of vibration:
If I understand correctly, the math related to atomic orbitals can be described with 3 dimensions of space: x, y and z plus one more orthogonal dimension of frequency/time which would mean quaternions would be most directly applicable?